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Apparent efficiency of serially coupled columns in isocratic and gradient elution
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modes
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AUTHORS: Szabolcs FEKETE1*, Santiago CODESIDO1, Serge RUDAZ1, Davy
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GUILLARME1, Krisztián HORVÁTH2
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1 School of Pharmaceutical Sciences, University of Geneva, University of Lausanne, CMU -
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Rue Michel Servet, 1, 1211 Geneva 4 – Switzerland
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2 Department of Analytical Chemistry, University of Pannonia, Egyetem u. 10, 8200
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Veszprém, Hungary
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CORRESPONDENCE: Szabolcs FEKETE
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Phone: +41 22 379 63 34
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E-mail: szabolcs.fekete@unige.ch
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*Manuscript
Click here to view linked References
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Apparent efficiency of serially coupled columns in isocratic and gradient elution
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modes
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Abstract
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The goal of this work was to understand the variation of apparent efficiency when serially
19
coupling columns with identical stationary phase chemistries, but with differences in their
20
kinetic performance. For this purpose, a mathematical treatment was developed both for
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isocratic and gradient modes to assess the change in plate numbers and peak widths when
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coupling arbitrary several columns. To validate the theory, experiments were also carried out
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using various mixtures of compounds, on columns packed with different particle sizes, to
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mimic highly efficient (new, not used) and poorly efficient columns (used one with many
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injections). Excellent agreement was found between measured and calculated peak widths.
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The average error in prediction was about 5 % (which may be explained by the additional
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volume of the coupling tubes).
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In isocratic mode, the plate numbers are not additive when the coupled columns possess
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different efficiencies, and a limiting plate count value can be calculated depending on the
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efficiency and length of the individual columns. Theoretical efficiency limit can also be
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determined assuming one column in the row with infinite efficiency.
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In gradient elution mode, the columns’ order has a role (non-symmetrical system). When the
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last column has high enough efficiency, the gradient band compression effect may
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outperform the competing band broadening caused by dispersive and diffusive processes
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(peak sharpening). Therefore, in gradient mode, the columns should generally be
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sequentially placed according to their increasing efficiency.
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Keywords:
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Column coupling, apparent efficiency, plate number, peak capacity, column length
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41
3
1. Introduction42
The idea of coupling columns to analyze complex samples appeared quite early in
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chromatography [1,2,3,4,5,6]. The purpose of column coupling can be either to improve
44
kinetic performance by increasing the column length or adjust selectivity by combining
45
different stationary phase chemistries. This latter idea lead to the development of
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multidimensional chromatographic separations.
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There are two ways to combine two or more columns in mono-dimensional separations,
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namely parallel and serial arrangements [7]. Serial columns generally outperform parallel
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setups, as the resolution power is appreciably extended in this configuration. The effect of
50
changes in column length is different in the serial and parallel approaches. The serially
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coupled columns approach has an intrinsic advantage: there is an additional separation
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factor (the column length), which however has no consequence in the experimental effort. In
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practice, each serial combination of short columns of different chemical nature and length
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operates as a new column, with its own selectivity. This increases enormously the wealth of
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columns available in a laboratory, from which the best one can be selected for a given
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application [7].
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In most cases, the aim of column coupling remains to increase the chromatographic
58
performance. The kinetic plot method (KPM) is often used as a design tool to find out the
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optimal column length to achieve a given number of theoretical plates [8,9]. The KPM can be
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used to predict the analysis time and efficiency which vary over a wide range of different
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column lengths, from very short to very long columns. Although the length independence is
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implicitly contained in the definition of the plate height concept, there are a number of cases
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wherein deviations from this behavior can be expected (axial temperature gradient due to
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viscous heating, extra-column band broadening effects which have relatively higher impact
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on small columns, side-wall effects that persist along the column length, pressure-related
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effects, etc.) [10,11,12,13]. The possibility to predict the performance of coupled columns
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systems has been extensively studied in the past. Coupling of up to six columns (900 mm = 6
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x 150 mm) showed that the KPM prediction was in good agreement with the obtained
69
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performance on the coupled column system [14]. In another study [15], it has been
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demonstrated that up to 8 columns (packed with 5 µm particles) could be coupled in series
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and operated at a constant flow rate without any significant loss of efficiency, again implying
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that the observed plate heights were independent on the column length.
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The best combination of coupled columns in isocratic mode having different lengths and
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particle sizes was determined in a previous study from Cabooter et al. based on the Knox-
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Saleem speed limit [16]. Considering an ultrahigh-performance liquid chromatography
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(UHPLC) system operating at a pressure of 1200 bar, the best possible serial connection
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system can approach the 75–85 % of its Knox-Saleem limit whereas a three-column parallel
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system can only get about 50–60 % of the speed limit, while needing 50–100 % more total
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column length. In absolute terms, the serially-connected system with individually optimized
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segment lengths should be able to cover a range of 5000–75,000 theoretical plates in an
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average analysis time of 14.3 min when using a 1200 bar instrument [16].
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When working in gradient mode, the overall peak capacity can be predicted in a very similar
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way on the basis of peak capacity measured on one single column, and assuming no
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differences in the performance of columns that will be coupled in series. Peak capacity
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prediction has indeed shown very good accuracy when coupling four columns of 150 mm in
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series [17]. Despite neglecting the possible variations in performance of the individual
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columns (different batches, history of the column), the kinetic performance limit approach
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works well in practice, as long as chromatographers couple the same type of columns (same
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stationary phase and dimension) in series.
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Therefore, in isocratic mode the plate numbers are expected to be additive, while in gradient
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mode, the peak capacity is proportional with the square root of the column length [18].
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Serial column coupling can be useful for various types of applications and is particularly used
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in RPLC mode [19]. By using a 450 mm long column (3 x 150 mm), the peak capacity of an
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antibody peptide mapping analysis was increased up to nc = 704 [20]. The same concept has
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also been used for intact and sub-units antibody analysis [20]. Another study showed the
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possibilities to achieve high plate count and peak capacity at various combinations of column
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lengths and temperatures [21]. Column coupling has also been applied in ion-exchange (IEX)
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chromatography to improve the separation of intact antibody charge-variants [22]. In
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supercritical fluid chromatography (SFC), 4 columns (4 x 100 mm) were successfully coupled
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to increase the separation of 24 pharmaceutical compounds [23]. Column coupling can also
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be applied for chiral separations [24]. As an example, an in-line coupling of achiral and chiral
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columns was shown to be a good alternative to multidimensional chiral chromatography [24].
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Coupling columns of different pore sizes in series is also commonly used in size exclusion
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chromatography (SEC) to tune the selectivity of polymer separations [25].
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Other purposes of column coupling can be post-column derivatization, on-line clean-up or the
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protection of the analytical column by using guard (pre-) columns [26,27].
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The purpose of this study was to estimate and measure the apparent efficiency of columns
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made of identical stationary phase chemistry but possessing differences in their kinetic
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performance. It may happen that the individual columns do not have identical efficiency
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(different batch, different lifetime and antecedents, or different packing quality which is well-
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known to be dependent on column length and diameter [28]). Different particle sizes were
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used to mimic columns of different batches or columns providing different efficiencies when
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coupling them in series. In isocratic mode, the plate numbers do not always seem additive
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and kinetic performance has a limiting value, depending on the efficiency and length of the
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individual columns. Furthermore, in gradient elution mode, the system is indifferent to the
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column order. Theory has been developed to show the evolution of plate numbers for
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coupling arbitrary several columns in isocratic mode and to predict peak widths for two
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columns system in gradient mode. Experimental measurements have been performed to
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validate the theory.
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2. Theory
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2.1 Peak widths in isocratic elution
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The band dispersion in serially connected columns can be calculated by solving the following
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ordinary differential equation:
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(1)
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where is the spatial variance of bands of compounds inside the column, the spatial
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variable, and the height equivalent to a theoretical plate, HETP.
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(2)
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where and are the HETP and length of the ith column, respectively. Note that is
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equal to zero.
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The solution of Eq. (1) in case of sequentially connected columns with the initial condition
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is:
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(3)
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Assuming that retention factors ( ) of solutes are the same in all the columns (identical
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stationary phase chemistry), the retention time of a compound can be expressed as:
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(4)
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where is the hold-up time, and is the average linear velocity of the eluent in the ith
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column.
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By matching the spatial (σz) and time (σ) variances through the definition of efficiency and
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replacing tR by Eq. (4), the following is obtained for a chromatographic peak eluted from
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sequentially connected columns is:
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(5)
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The fraction on the right hand side of Eq. (5) can be expressed from Eq. (4) as:
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(6)
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where Ln and are the length and dead volume of the last segment, is the total dead
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volume of the sequentially connected columns. Explicitly, with and are
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the internal diameter and total porosity of column .
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Eq. (6) can be combined with Eq. (5) and the peak width can be calculated as:
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(7)
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where,
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(8)
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The total plate number of the sequentially connected columns is the sum of the number of
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theoretical plates of the columns. The apparent plate number, however, can be calculated
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as:(9)
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Eqs. (8) and (9) can be generalized after the following considerations:
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, ,
(10)
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Accordingly,
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(11)
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and,
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(12)
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The ratio of and the total plate number is:
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(13)
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For a two-column system Eqs. (11), (12) and (13) become:
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(14)
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(15)
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(16)
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There are also several specific situations for two-columns:
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If the plate numbers of the two columns are equal:
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(17)
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If the column diameters are equal:
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(18)
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If the column diameters and lengths are equal:
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(19)
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If the plate numbers and diameters of the two columns are equal:
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(20)
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If the efficiency of one of the two columns is infinite (N2= ∞)
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(21)
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If the efficiency of one of the two columns is infinite (N2= ∞)) and the column dimensions are
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identical:
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(22)
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Peak capacity, , can be obtained as the solution of the following ordinary differential
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equation with the initial condition of :
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(23)
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where is peak width as a function of time, .
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The peak capacity of a series of columns connected together in isocratic mode can then be
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calculated as the solution of Eq. (23):
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(24)
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For a two-columns system, the following equation can be written:
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(25)
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If the column dimensions are identical and one of the columns has an infinite efficiency (N2 =
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∞):(26)
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2.2 Peak widths in gradient elution
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In gradient chromatography, the arrival time of a peak to a position z along the column is no
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longer just proportional to time, but has to be retrieved from solving a differential equation,
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given by:
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(27)
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where the velocity u is given by the instantaneous linear velocity of the solute, related to that
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of the mobile phase (u0) through the solute retention (k):
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(28)
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Notice that the condition implies a negligible dwell volume. This is always the case
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for initially highly retained compounds, that are stopped at the head of the column until the
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gradient releases them. According to the linear solvent strength (LSS) theory, the retention
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factor can be written as a function of the mobile phase composition [29]:
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(29)
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where S is the slope of the LSS model (log k vs. % organic modifier) and kw is the
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extrapolated value of k for a compound eluted with pure A eluent (i.e., Φ=0). When running a
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linear gradient over a time the mobile phase composition at the inlet of the column is given
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by:
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(30)
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The retention at a time t and position z, taking into account the time it takes for the
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mobile phase to reach that point, will be (again neglecting the dwell volume):
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(31)
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where is the initial retention, L the length of the column, t0 = L/u0 the hold-up
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time, and:
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(32)
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Is the intrinsic gradient steepness.
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The solution of (27) is the well-known chromatography formula:
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(33)
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The time to travel to z = L is the retention time, expressed as:
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(34)
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Peak width is mostly affected by diffusion and dispersion processes and by the gradient band
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compression effect. The peak is compressed because of the changes to its trajectory while
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crossing the gradient within the column. During gradient elution, the rear part of the peak
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moves faster than its front part, because the mobile phase strength increases along the
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column. The steeper the gradient, the higher the band compression effect is. To model the
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band compression effect, it is useful to consider a peak between a point z and = z + .
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Then the next formula can be written:
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(35)
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When w is small compared to the total length over which the motion of the peak is integrated,
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the right-hand side of eq (35) can be expanded at first order in w to obtain:
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(36)
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Our goal is to have z as an independent variable, so with the chain rule, the following
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equation can be obtained:
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(37)
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The speed gradient at a given position is obtained by plugging in the solution the equation
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(33):
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(38)
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where,
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(39)
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is a measure of the gradient steepness. It takes into account that initially unretained
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substances (k0 = 0) will not be compressed at all.
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The band broadening effects can be dependent on the column HETP measured in isocratic
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mode, which we call here H0,
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(40)
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Equation (40) can be joined with (37) to give:
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(41)
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If the width at a given point along the column z0 (this will be necessary for the coupling)
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is known:
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(42)
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the solution is:
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(42)
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By neglecting the initial peak width caused by the injection process, when z = 0, we
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obtain the known formula at elution (z = L):
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(43)
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We now assume a two-column system possessing different HETP values, H1 for length L1,
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and H2 for length L2. If the same gradient steepness and linear velocity are considered on the
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two columns, then the migration can still be described by equation (33), with L = L1 + L2. In
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the first column, the width evolves from the injection width at z = 0. By setting H0 = H1,
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= and z0 = 0 in equation (42), the following equation can be obtained:
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(44)
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When it reaches z = L1, it starts migrating under H2. The coupling condition is:
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(45)
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This means that follows equation (42), with H0 = H2, z0 = L1 and = (L1), thus
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defining:
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(46)
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the solution becomes:
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(47)
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or over the whole coupled system:
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(48)
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The result is very similar to (44), with a correction factor proportional to . At elution, z = L =
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L1 + L2 :
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(49)
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Please note that the dependence in H1 only comes through . In particular, if L1 is smaller
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than L, then the efficiency is basically dominated by the second column.
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3. Experimental
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3.1 Chemicals and columns
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Acetonitrile, methanol and ethanol (gradient grade) were purchased from Sigma-Aldrich
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(Buchs, Switzerland). Water was obtained with a Milli-Q Purification System from Millipore
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(Bedford, MA, USA).
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Uracil, methylparaben, ethylparaben, propylparaben, butylparaben, cannabidivarine (CBDV),
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cannabigerolic acid (CBGA), tetrahydrocannabivarin (THCV), cannabichromene (CBC),
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delta9-tetrahydrocannabinolic acid (THCA-A) and human serum albumin (HSA), were
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purchased from Sigma–Aldrich. Cannabidiolic acid (CBDA), cannabigerol (CBG), cannabidiol
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(CBD), cannabinol (CBN), (-)-delta9-THC (d9-THC) and (-)-delta8-THC (d8-THC) were
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purchased from Lipomed AG (Arlesheim, Switzerland).
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Ammonium hydroxide solution, formic acid (FA), trifluoroacetic acid (TFA), dithiothreitol
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(DTT) and trypsin were obtained from Sigma-Aldrich.
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X-Bridge C18 (5 µm, 150 x 4.6 mm) (A), X-Bridge C18 (3.5 µm, 100 x 4.6 mm) (B) and X-
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Bridge C18 (2.5 µm, 75 x 4.6 mm) (C) columns were purchased from Waters (Milford, MA,
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USA). Jupiter C18 (5 µm 300 Å, 150 x 2.0 mm) (D) and Jupiter C18 (3 µm 300 Å, 150 x 2.0
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mm) (E) columns were purchased from Phenomenex (Torrance, CA, USA).
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3.2 Equipment and software
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The measurements were performed using a Waters Acquity UPLCTM I-Class system
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equipped with a binary solvent delivery pump, an autosampler and UV detector and/or
300
fluorescence detector (FL). The system includes a flow through needle (FTN) injection
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system equipped with 15 µL needle, a 0.5 µL UV flow-cell and a 2 µL FL flowcell. The
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connection tube between the injector and column inlet was 0.003” I.D. and 200 mm long
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(active preheating included), and the capillary located between the column and detector was
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0.004” I.D. and 200 mm long. The overall extra-column volume (Vext) was about 8.5 µL and
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11 µL as measured from the injection seat of the auto-sampler to the detector cell (UV and
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FL, respectively). The average extra-column peak variance of our system was found to be
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around ∼0.5 − 3 µL2 (depending on the flow rate, injected volume, mobile phase
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composition and solute). Data acquisition and instrument control were performed by
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Empower Pro 3 Software (Waters).
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3.3 Chromatographic conditions and sample preparation
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3.3.1. Apparent plate numbers: Isocratic measurements of parabens and uracil
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A mix solution containing uracil, methylparaben, ethylparaben, propylparaben and
314
butylparaben was prepared in 80 : 20 v/v water : acetonitrile at 50 µg/mL.
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For isocratic chromatographic measurements, the mobile phase was composed of 55 : 45 v/v
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water : acetonitrile. Experiments were performed at a flow rate of 1 mL/min at ambient
317
temperature. Detection was carried out at 254 nm (40 Hz), the injection volume was 5 µL.
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The plate numbers were measured on three single columns, namely the 5 µm, 150 x 4.6 mm
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(A), 3.5 µm, 100 x 4.6 mm (B) and 2.5 µm, 75 x 4.6 mm (C) columns, then the columns were
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coupled in series using 5 cm long (0.175 mm ID) stainless steel tubing and the apparent
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plate numbers were measured. The following combinations were tested: (1) columns A + B,
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(2) columns A + C, (3) columns B + C and (4) columns A + B + C.
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3.3.2. Apparent peak widths: gradient measurements of small molecules (mix of
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cannabinoids)
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A mix solution containing eleven cannabinoids (i.e. CBDV, CBGA, THCV, CBC, THCA-A,
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CBDA, CBG, CBD, CBN, d9-THC and d8-THC) was prepared from individual stock solutions
328
diluted in solvent having the same composition as the initial mobile phase (55 : 45 v/v 10 mM
329
ammonium-acetate : acetonitrile) at 90 µg/mL. The individual stock solutions were prepared
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in either methanol, acetonitrile or ethanol depending on their solubility. Mobile phase “A” was
331
10 mM ammonium-acetate (pH = 5.8), mobile phase “B” was acetonitrile. Linear gradients
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were run from 45 %B to 100 %B at 1 mL/min flow rate and ambient temperature. The
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gradient time (tG) over column length (L) ratio was kept constant (tG/L=1 min/cm) when
334
running gradients on different column lengths (corresponds to e.g. tG = 10 min on 10 cm long
335
column). Detection was carried out at 220 and 254 nm (40 Hz), the injection volume was 10
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µL. The peak widths (peak capacity) were measured on three single columns, namely on the
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5 µm, 150 x 4.6 mm (A), 3.5 µm, 100 x 4.6 mm (B) and 2.5 µm, 75 x 4.6 mm (C) columns,
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and then these columns were coupled in series using 5 cm long (0.175 mm ID) stainless
339
steel tubing, and the apparent peak widths (peak capacity) were measured. The following
340
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combinations were used: (1) columns A + B, (2) columns A + C, (3) columns B + C and (4)
341
columns A + B + C.
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3.3.3. Apparent peak widths: gradient measurements of peptides (HSA tryptic digest)
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Tryptic digestion of human serum albumin (HSA) was carried out as described in a recent
345
protocol [30]. Mobile phase “A” was 0.1 % TFA in water, mobile phase “B” was 0.1 % TFA in
346
acetonitrile. Linear gradients were run from 10 to 70 %B at a flow rate of 0.3 mL/min and 50
347
˚C. The gradient time (tG) over column length (L) ratio was kept constant (tG/L=2 min/cm)
348
when running gradients on different column lengths (corresponds to e.g. tG = 30 min on 15
349
cm long column). Fluorescence detection was carried out at 280 nm as excitation and 350
350
nm as emission wavelengths, the injection volume was 5 µL. The peak widths (peak
351
capacity) were measured on two widepore columns of 5 µm, 150 x 2.0 mm (D) and 3 µm,
352
150 x 2.0 mm (E) columns, then these two columns were coupled in series using 5 cm long
353
(0.175 mm ID) stainless steel tubing, and the apparent peak widths (peak capacity) were
354
measured. The following combinations were used: (1) columns D + E and (2) columns E + D.
355 356
4. Results and Discussion
357
4.1. Apparent plate number in isocratic mode for serially coupled columns
358
As it is possible to couple together two columns possessing different lengths and efficiencies,
359
an informative representation of the apparent plate number (Napp) can be obtained when
360
plotting the ratio of Napp/Nsum (where Nsum is the sum of the individual plate counts) as a
361
function of N1/N2 (corresponding to the efficiency of the first and second columns,
362
respectively). In this type of representation, various ratios of column lengths (L1/L2) can be
363
tested. Figure 1 shows some plots for L1/L2 = 0.75, 1, 1.5 and 2 (calculations are based on
364
eq 18). As suggested by the theory, all the curves show a maxima (Napp/Nsum = 1), indicating
365
that the highest reachable efficiency with two serially coupled columns is equal to the sum of
366
the individual plate numbers. However, it only occurs at a given ratio of column lengths.
367
When coupling two columns of identical lengths (L1/L2 = 1) in series, then this maximum
368
16
occurs when the columns possess identical plate numbers (N1/N2 = 1). When the first column
369
is twice longer than the second one (L1/L2 = 2), then the maximum plate number is attained
370
when the first column performs twice as high plate numbers than the second one (N1/N2 = 2).
371
Similarly, when L1/L2 = 1.5 and 0.75, the maximum performance is expected for N1/N2 = 1.5
372
and 0.75, respectively. Accordingly, to obtain the maximum efficiency from equal coupled
373
columns configuration, it is required that their plate heights should be the same. In this case,
374
the apparent plate count is the sum of the individual plate counts. In any other case,
375
Napp/Nsum will be smaller than 1. An important feature of the system is its symmetric property,
376
meaning that the system (or N) is indifferent to the order. One can choose the more efficient
377
column either in the first or the second position, without affecting the global efficiency.
378
Table 1 shows the measured and calculated plate numbers on single columns (A, B and C)
379
and serially coupled configurations (including two and three columns) for four model
380
compounds (parabens). As shown, the measured and predicted plate numbers are in very
381
good agreement, with a variation between measured and predicted efficiency comprised
382
between -7 and +5 %. Figure 1 also includes the experimentally measured values which fit
383
quite well with the theoretical curves. As an example, Figure 2 shows some representative
384
chromatograms of the four parabens separated on three different columns with different
385
lengths and efficiency as well as with the three columns serially coupled.
386
Another interesting aspect is to track the efficiency increase of two serially coupled columns
387
compared to just one of the columns used for this coupling. Figure 3 illustrates Napp/N1 as a
388
function of N1/N2 for three cases, namely for L1/L2 = 0.2, 1 and 2 with identical column
389
diameters. When L1/L2 = 2 (the first column is twice as long as the second one), the intercept
390
of the curve corresponds to Napp/N1 = 2.25. This means that the maximum efficiency is 2.25
391
times higher vs. that of the first column. It occurs when the second column has infinite
392
efficiency (intercept at N1/N2=0). In this case, the second column only increases retention
393
times without any effect on band broadening. As illustrated in Figure 3, it is not possible to
394
attain higher plate numbers with this setup. On the other hand, if the efficiency of the second
395
column is five times lower than that of the first column, then the apparent plate number of
396
17
serially coupled columns will be the same as of the first column. When N1/N2 is above 5, the
397
overall efficiency of the coupled system is lower than the efficiency of the first column. This
398
means that it is possible to combine two HPLC columns that finally generate lower resolution
399
than that offered by the most efficient column alone. This counter instinctive consequence is
400
analogous to the band broadening effect due to the extra column contributions. Similarly,
401
when L1/L2 = 1, the maximum achievable efficiency is four times higher than that of the first
402
column, while if the efficiency of the second column is at least three times lower than the first
403
column, then no increase in efficiency is obtained when coupling these two columns. Finally,
404
when the first column is very short compared to the second column (L1/L2 = 0.2) and the
405
second column has very high efficiency (infinite) then Napp/N1 = 36 can be attained when
406
coupling the columns.
407
In general, when efficiency of the second column is infinite, the apparent plate number of the
408
two-column system becomes:
409
(50)
410
The condition when additional gain of efficiency can be obtained by coupling two columns is:
411
(51)
412
or similarly,
413
(52)
414
where , , and .
415
Accordingly, additional gain of efficiency and resolution is possible by coupling two HPLC
416
columns only if the column plate heights do not differ too significantly.
417 418
4.2. The evolution of peak width and peak capacity in gradient mode for serially coupled
419
columns
420
In gradient elution mode, the order of the columns is concerned, and the observed apparent
421
efficiency strongly depends on the order of the columns (non-symmetrical system). An
422
illustration is given in Figure 4. Assuming two columns (with the same internal diameter) with
423
18
plate heights, H = 10 μm and H = 40 μm, respectively coupled in series. The peak width will
424
evolve in different ways depending on the column order and length of the individual
425
segments (the different plate heights were assumed to mimic columns of different batches or
426
the combination of old and new columns). The continuous lines in Figure 4 show the peak
427
widths for coupled columns possessing different efficiencies as a function of the position of
428
the solute (z) along its travel. The dashed lines correspond to columns having either H = 10
429
μm or H = 40 μm efficiency along its entire length (10 cm) – as reference values.
430
Figure 4A shows the case where two segments of 5 cm are coupled at a moderate gradient
431
steepness (p=1). When placing the more efficient column in the first position and the less
432
efficient one in the second position (continuous red line) – as expected – the peak will
433
broaden drastically after entering the second (less efficient) column as the band broadening
434
caused by dispersion and diffusion processes becomes more important. However, when
435
having the less performing column in the first position and the most efficient column in the
436
second position (continuous blue line), interestingly the peak width will decrease
437
continuously during the travel of the solute along the second column (“peak sharpening”). It
438
suggests that the gradient band compression effect outperforms the dispersive and diffusive
439
effects in the second column as the more efficient column offers much lower H value than the
440
first column. If the second - more efficient - column is very long compared to the first one, the
441
peak width will approach the limiting value theoretically obtained only with the more efficient
442
column - indeed, the dashed line (single column with maximal efficiency) is an asymptote of
443
the solid line (coupled system), that are equal in the large z limit.
444
Figure 4B represents a situation where the column lengths are different. The first one is four
445
times shorter than the second one. When placing the better column in the first position, then
446
a trend similar to that of Figure 4A can be seen. However the coupled system approaches
447
faster its limit (see the dashed and continuous red lines) because at the beginning of the
448
solute’s travel along the column, the gradient compression effect is stronger than later during
449
the travel (e.q. 38). When putting the more efficient column as the second one, then no band
450
broadening occurs in the second column, and the peak width remains more or less constant
451
19
whilst the solute is traveling through the second column (continuous blue line). It suggests
452
that the gradient band compression effect nearly compensates the band broadening caused
453
by dispersion and diffusion processes. Please note that the differences between the coupled
454
systems – with columns possessing different efficiencies - were larger in this case compared
455
to the situation where the lengths were identical (see the differences at z = 10 cm between
456
the continuous blue and red lines in Figures 4A and 4B).
457
Finally, figure 4C corresponds to a situation with two columns of 5 cm – similarly to Figure 4A
458
– but for a steeper gradient (p = 10). The trends were similar as the ones observed in Figure
459
4A, but as expected the gradient focusing effect was more important, and therefore the total
460
peak width was smaller. When placing the better column in the second position (continuous
461
blue line), the speed of peak compression was faster on the second column compared to the
462
case where a flatter gradient was applied.
463
To verify the theory developed for predicting the peak width in gradient mode, two sets of
464
compounds were analyzed using serially connected columns having different particle sizes
465
and lengths. Figure 5 shows the separations of 11 cannabinoids on three different individual
466
columns and on different combinations of two or three columns, as selected examples. Table
467
1 contains the experimentally measured and predicted peak widths for the first and last
468
eluting peaks. The peak width prediction for serially coupled columns was based on the peak
469
widths measured on the individual columns. In particular, values for single column
470
efficiencies were retrieved from direct measurements of peak width. These efficiencies were
471
then used as the input for the coupled formula (e.q. 49). The measured and calculated widths
472
were in very good agreement, as the average error in prediction was about 5-6 %.
473
Another experimental verification was performed by injecting HSA tryptic digest on two
474
individual widepore columns packed with 5 and 3 μm particles and on the combination of
475
these two columns in different orders (Figure 6). Larger molecules (peptides) possess higher
476
S values, therefore it was interesting to check the validity of the model calculations for such
477
molecules. The peak widths of the three most intense (and well separated) peaks was
478
predicted for the coupled systems from the widths on the single columns. Again, very good
479
20
agreement was found between experimentally observed and calculated peak widths (Table
480
3), as the average error in prediction was about 6 %. The results confirm the importance of
481
the columns order as the order “D + E” always gave thinner peaks than “E + D” (both for
482
predicted and measured peak widths).
483 484
5. Conclusions
485
The serially coupled columns approach has an intrinsic advantage as it offers an additional
486
separation factor (the column length). In most cases, the column length is increased by
487
coupling columns packed with the same material (i.e. stationary phase and particle size). In
488
this case, the plate number observed with the coupled column system is the sum of the plate
489
counts observed on the individual column segments. However, it may happen that the
490
individual columns do not have identical efficiency (different batch, different lifetime and
491
antecedents, or different packing quality which is well-known to be dependent on column
492
length and diameter). Therefore, coupling columns with different efficiencies in series raises
493
some questions: (1) What will be the final apparent efficiency?, (2) What is the maximum
494
efficiency that can be reached?, and (3) Does the column order play a significant role?
495
Theory was developed for both isocratic and gradient modes, to predict the peak widths for
496
coupled column systems. In isocratic mode, the plate numbers are not additive anymore
497
when the columns possess different plate count, and kinetic performance has a limiting value
498
which depends on the efficiency and length of the individual columns.
499
Furthermore, in gradient elution mode, the order of the columns is not indifferent. Indeed, the
500
observed apparent efficiency significantly depends on the column order (non-symmetrical
501
system). In combinations, when the latter column has higher efficiency, a decrease of the
502
peak width is predicted (“peak sharpening”), when the solute travels this segment. This
503
means that the gradient band compression effect compensates and outperforms the
504
competing band broadening caused by dispersive and diffusive processes. Therefore, the
505
columns should be placed in order of increasing efficiency.
506
21
Experimental measurements have been performed in both isocratic and gradient modes to
507
verify the developed theory. Very good agreement was found between measured and
508
calculated peak widths.
509
To conclude for serially coupled column systems in gradient mode, besides the total length of
510
the coupled column, additional important factors are the order and lengths of the individual
511
segments which must be considered when optimizing a gradient separation.
512 513
6. Acknowledgements
514
The authors wish to thank Jean-Luc Veuthey and Balazs Bobaly from the University of
515
Geneva for fruitful discussions.
516
Davy Guillarme wishes to thank the Swiss National Science Foundation for support through a
517
fellowship to Szabolcs Fekete (31003A 159494).
518
Krisztián Horváth acknowledges the financial support of the Hungarian Government and the
519
European Union, with the co-funding of the European Social Fund in the frame of GINOP
520
Programme [Code No: GINOP-2.3.2-15-2016-00016], and of the János Bolyai Research
521
Scholarship of the Hungarian Academy of Sciences.
522
22
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the peak capacity using coupled columns packed with 2.6 µm core–shell particles operated
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particles size and ultra-high pressure, J. Chromatogr. A 1216 (2009) 3232-3243.
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by human liver microsomes, J. Chromatogr. A 1467 (2016) 326-334.
594
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599
Anal. Chim. Acta, 889 (2015) 58-70.
600
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coupling with HPLC, J. Chromatogr. B, 911 (2012) 49-54.
603
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604
disposable pre-packed columns for protein chromatography, J. Chromatogr. A, 1527 (2017)
605
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[29] L.R. Snyder, J.W. Dolan, High-performance gradient elution: The practical application of
607
the linear solvent strength model, John Wiley & Sons, Inc. 2007
608
[30] B. Bobaly, V. D’Atri, A. Goyon, O. Colas, A. Beck, S. Fekete, D. Guillarme, Protocols for
609
the analytical characterization of therapeutic monoclonal antibodies. II – Enzymatic and
610
chemical sample preparation, J. Chromatogr. B 1060 (2017) 325-335.
611
612
613
26
Table 1.614 615
column L (mm)
N
methylparaben ethylparaben propylparaben butylparaben measured predicted measured predicte
d measured predicted measured predicted
A 150 15111 - 14911 - 14763 - 15070 -
B 100 6066 - 6329 - 6512 - 6889 -
C 75 11509 - 12046 - 12478 - 12681 -
A+B 250 19388 19920 19792 20233 19863 20427 20956 21225
A+C 225 24744 25598 24955 25621 25599 25635 25902 26414
B+C 175 14406 14329 15301 14961 15921 15417 16440 16160
A+B+C 325 27857 29128 28757 29704 29256 30088 30690 31174
616
27
Table 2.617 618
column L (mm)
peak 1 peak 11
w1/2
measured (min)
w1/2
predicted (min)
w1/2
measured (min)
w1/2
predicted (min)
Rs crit 9.10 peak capacity
A 150 0.0571 - 0.0715 - 0.69 134
B 100 0.0666 - 0.0771 - 0.55 83
C 75 0.0375 - 0.0399 - 0.57 119
B+A 250 0.0909 0.0993 0.1031 0.1162 0.88 144
C+A 225 0.075 0.0698 0.0838 0.0870 0.99 159
C+B 175 0.0852 0.0861 0.0964 0.0994 0.57 110
A+B+C 325 0.1054 0.0957 0.1098 0.1084 1.00 167
619
620
28
Table 3.621 622
column L (mm)
peak 1 peak 2 peak 3
w1/2
measured (min)
w1/2
predicted (min)
w1/2
measured (min)
w1/2
predicted (min)
w1/2
measured (min)
w1/2
predicted (min)
peak capacity
D 150 0.0697 - 0.0703 - 0.0781 - 240
E 150 0.0592 - 0.0559 - 0.0525 - 312
D + E 300 0.0918 0.0857 0.0849 0.0819 0.088 0.0797 388
E + D 300 0.0923 0.0968 0.095 0.0971 0.0966 0.1066 362
623
624
29
Figure captions625 626
Figure 1. Napp/Nsum (relative apparent efficiency of the coupled system) as a function of N1/N2
627
(ratio of individual column efficiency) for various column length ratios (L1/L2 = 0.75, 1, 1.5 and
628
2).
629 630
Figure 2. Experimentally obtained chromatograms of a mixture of uracil and 4 parabens on
631
columns A, B and C and on serially connected columns A+B+C in isocratic mode. The
632
mobile phase was composed of 55 : 45 v/v water : acetonitrile. Experiments were performed
633
at a flow rate of 1 mL/min at ambient temperature. Detection was carried out at 254 nm (40
634
Hz), the injection volume was 5 µL. Peaks: uracil (t0), methylparaben (1), ethylparaben (2),
635
propylparaben (3) and butylparaben (4).
636 637
Figure 3. Napp/N1 (apparent efficiency compared to the first column) as a function of N1/N2
638
(ratio of individual column efficiency) for various column length ratios (L1/L2 = 0.2, 1, and 2).
639 640
Figure 4. The evolution of peak variance ( along the column (z) for a system composed of
641
two columns coupled in series. Three cases named A to C are reported, corresponding to
642
different segment lengths and gradient steepness, considering H = 10 µm and 40 μm. Please
643
note that time based peak width as a practical measure of band broadening can be obtained
644
at the total length by .
645
646
Figure 5. Experimental chromatograms of cannabinoids mixture on columns A, B and C and
647
on serially connected combinations in gradient mode. Linear gradients were run from 45 to
648
100 %B at 1 mL/min and ambient temperature. The gradient time (tG) over column length (L)
649
ratio was kept constant (tG/L=1 min/cm) when running gradients on different column lengths.
650
Detection was carried out at 254 nm (40 Hz), and injection volume was 10 µL.
651 652
Figure 6. Experimental chromatograms of HSA tryptic digest on columns D and E and on
653
serially connected “D+E” and “E+D” combinations in gradient mode. Linear gradients were
654
run from 10 to 70 %B at 0.3 mL/min and 50 ˚C. The gradient time (tG) over column length (L)
655
ratio was kept constant (tG/L=2 min/cm) when running gradients on different column lengths.
656
Detection was carried out at 280 nm as fluorescence excitation and 350 nm as fluorescence
657
emission wavelengths, and injection volume was 5 µL.
658
659
660
30
Table captions661 662
Table 1. Measured and predicted plate numbers for parabens in isocratic mode on three
663
individual columns and on their different combinations. (Predictions are based on eq. 18 for
664
two columns and 12 for three columns.)
665
666
Table 2. Measured and predicted peak widths for cannabinoids in gradient elution mode on
667
three individual columns and on their different combinations. The obtained critical resolution
668
and peak capacity are also shown. (Predictions are based on eq. 47-49.)
669
670
Table 3. Measured and predicted peak widths for peptides obtained in gradient elution mode
671
on two individual columns and on their combinations. The obtained peak capacity is also
672
indicated. (Predictions are based on eq. 47-49.)
673
674
Figure 1
Figure